Distribution compaction aims to accurately summarize the probability distribution $\mathbb{P}$ using a small number of representative points. A near-optimal decimation procedure samples $n$ points from a Markov chain and identifies $\widetilde{\mathcal{O}}(1/\sqrt{n})$ discrepancies at $\sqrt{n}$ points. achieve this goal by to $\mathbb{P}$. Unfortunately, these algorithms suffer from quadratic or superquadratic runtimes with sample size $n$. To address this shortcoming, we introduce Compress++. This is a simple meta-step to speed up the decimation algorithm, while giving an error of at most $4$. Combining the quadratic-time kernel halving and kernel decimation algorithms of Dwivedi and Mackey (2021), Compress++ achieves $\sqrt{n}$ points in $\mathcal{O}(\sqrt{\log n/n})$ provide. Integration error in $\mathcal{O}(n \log^3 n)$ time and $\mathcal{O}( \sqrt{n} \log^2 n )$ and maximum mean discrepancy space better than Monte Carlo. In addition, Compress++ enjoys the same nearly linear execution time given any quadratic time input, reducing the execution time of hyperquadratic algorithms by a square root factor. In benchmarks using Markov chains targeting high-dimensional Monte Carlo samples and difficult differential equation posterior distributions, Compress++ matches or nearly matches the accuracy of the input algorithm in orders of magnitude faster.
